Interpretation of the Near Field
Hi,
I would like to know how we can relate near field solutions and reflection coefficients in the simulations of periodic structures. When I simulate the Jerusalem cross FSS model in the example guide, near field results above the structure are always greater than the incident field (which is 1 V/m). Moreover, when the reflection coefficients are requested on the plane positioned above the structure where near field results are calculated, there is no correlation between near field results and reflection coefficient results when they are plotted as a function of frequency. Since the reflection coefficient is basically Er/Ei, how should we interpret near field results and Er?
Thanks.
Answers
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Hi hykr
By default FEKO would calculate the total field for a near field request. As such, what you are seeing above the structure would be a combination of the incident and reflected fields: Etotal = Ei + Er. This total field could be expected to be higher than the incident field in some locations (where the incident and reflected fields add in phase).
If you are interested in calculating only the reflected near field, you could select the 'Calculate only the scattered part of the field' option under the Advanced tab of the near field request.
Regards,
Johan H
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Hello hykr,
I suspect that you are simply placing your near filed request points too close to the FSS. Before FEKO had the transmission and reflection coefficient requests, these values were calculated from the near fields. I remember the example calculated it at a distance of 10 wavelengths at the lowest frequency. If you are too close, you see an interference pattern (oscillation). You should also remember to only request the scattered field calculation in FEKO - by default it will calculate the total field (incident + scattered).
It could be wroth your time to read the following (extracted from an old article that we had on the website - will be added to the manual):
Understanding Grating Lobes in the Context of Periodic Boundary Conditions
Simple equations are presented to help understand the appearance of grating lobes in connection with PBC simulations in FEKO.The purpose of this article is to discuss grating lobes in connection with periodic boundary conditions (PBC) in FEKO. Simple equations are derived to determine the frequency where grating lobes start to form, as well as the grating lobe observation angles in the far-field. These equations are verified by means of a Jerusalem-cross frequency selective surface (FSS) example. The effect of grating lobes on the computation of the reflection and transmission coefficients will also be shown.
In order to derive these equations, consider the infinite one dimensional array shown in Figure 1 (similar for 2-D). The element spacing is d and the incident plane wave excitation is at an angle θi. We are interested in the far-field at the observation angle θ. The PBC assumes that the current distributions on all array elements are identical (except for the phase increment). To compute the far-field we sum the contribution of all the elements to obtain:
with the wave number k=2π/λ, the phase term above is:
Figure 1: Infinite One Dimensional ArrayAll elements will radiate in phase when m = 0, 1, 2 for:
Therefore, in the far-field we will have in phase radiation at the observation angles:
For each solution θ there is a corresponding solution 180° - θ, since sin(180° - θ) = sin(θ).
Looking at this last equation, some important properties can be seen:
- The main beams correspond to m=0 at the observation angles θ = θi and θ = 180° - θi.
- The first grating lobe correspond to m=1. It will only exist for real angles θ, when the magnitude of the argument of arcsin is smaller than unity.
- No grating lobes exist if the element spacing is smaller than d ≤ 0.5 λ (for all incident angles θi)
- As the spacing d is increased in terms of wavelength λ, grating lobes will start to form along the end fire directions at θ = ± 90°.
Two more equations can be derived from the above equations at the onset of grating lobes.
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The incident angle where grating lobes start from [1, eq. 8]
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The frequency where grating lobes start from [2, eq. 14]
where:
- m = 1, 2, etc. (m = 0 gives no grating lobes)
- θi is the incident plane wave excitation angle
- d is the element spacing in the periodic array
- c is the speed of light
Example: Jerusalem Cross with PBC Unit Cell
Consider the Jerusalem-cross FSS [3] with the unit cell shown in Figure 2. This example is used to verify the predicted frequency where grating lobes start to form, as well as the predicted observation angles for the main beams and grating lobes in the far-field. The influence of grating lobes on the computation of the reflection and transmission coefficients will also be shown.
Figure 2: Jerusalem cross FSS with PBC unit cell
Parameters of interest: plane wave excitation in the xz-plane with incident angle θi=30°, and square 2-D periodic lattice with element spacing d = 15.2 mm. The dimensions of the FSS can be found in [3].
The first predicted frequency where grating lobes start to form is: f = c / d / (1 + sin(θi)) = 13.2 GHz.
Below this frequency there will be no grating lobes, only the two main beams at θ = 30°,150° (they are always present at all frequencies). Above this frequency grating lobes start to form in addition to the two main beams, i.e.
- At f = 13.5 GHz there are two grating lobes at the predicted angles θ = -73.9°, 253.9°.
- At f = 14.0 GHz there are two grating lobes at the predicted angles θ = -65.3°, 245.3°.
To validate the predicted observation angles for the main beams and grating lobes, the far-field radiation pattern of a large finite 51x51 FSS array was analyzed in FEKO using the PBC. This was done at the frequencies f = 13.0 GHz, 13.5 GHz and 14.0 GHz as shown in Figures 3-5. A cut in the xz-plane of the far-field pattern is superimposed on the FSS unit cell in all these figures. The predicted main beam angles are identical to those computed at θ = 30°,150°. Predicted grating lobe angles are within 0.6° at 13.5 GHz and within 0.3° at 14.0 GHz of the computed angles.
Figure 3: Far-field at 13.0 GHz (no grating lobes)
Figure 4: Far-field at 13.5 GHz, computed grating lobes at θ = -73.9°, 253.9°.
Figure 5: Far-field at 14.0 GHz, computed grating lobes at θ = -65.3°, 245.3°.
To determine the reflection and transmission coefficients of the FSS, the near electric fields are computed at a large distance away from the FSS (i.e. at ten wavelengths). The FSS in Figure 2 is located at z=0 and the reflection coefficient is determined from the scattered near-field computed along the positive z-axis. Similarly, the transmission coefficient is determined from the total near-field computed along the negative z-axis. In the absence of grating lobes, only the main beam contributes to the near-fields. However, as the frequency is increased, grating lobes will form that will also contribute to the near-fields. For an infinite 2-D periodic structure these main beams and grating lobes are propagating plane waves at the observation angles θ (predicted by our simple equation). The effect of grating lobes on the computation of the transmission coefficient can be seen in Figure 6. Here the near-fields below (f = 13.0 GHz) and at the onset (f = 13.2 GHz) of grating lobes is computed along the negative z-axis. At 13.0 GHz only a single propagating plane wave exists (corresponding to the main beam) and the amplitude becomes constant for large z values. At 13.2 GHz there will be two propagating plane waves in the lower half space (corresponding to the main beam and the grating lobe). The interference between these two plane waves causes the oscillation in the amplitude of the near-field, also for large z distances. So when grating lobes exist, one cannot simply compute the reflection or transmission coefficients by looking at the near-fields at a single point. This procedure is only valid if a single propagating plane wave exists, i.e. for frequencies below 13.2 GHz.
Figure 6: Near-field along the negative z-axis
Figure 7 shows the computed reflection and transmission coefficients of the FSS versus frequency. The near-fields are computed at two single observation points at z = ±0.1 m. As mentioned previously, these results are only valid for frequencies up to 13.2 GHz (before the onset of grating lobes) where a small spike can be seen. The rapid variation at 9.4 GHz is correct (this is not a grating lobe) and was verified by using the multilevel-fast-multipole method to analyze the complete finite array. Also shown in Figure 8 is the reflection coefficient for a normal incident plane wave. Good agreement between FEKO and the published results can be seen. The PBC analyses show that the performance of the Jerusalem-cross FSS degrades for θi =30° incidence around 9.4 GHz.
Figure 7: Reflection and transmission coefficients for θi = 30° (only valid below 13.2 GHz)
Figure 8: Reflection coefficient [3] for θi = 0°
Please let me know if you see the expected correlation after considering the comments above.
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Hi JIF,
Thank you for your response. As Johan suggested, selecting the 'Calculate only the scattered part of the field' option yields only the reflected field, thus its amplitude is in accordance with the reflection coefficient. However I am more interested in finding the reflection phase but the phase of the reflected field and the phase of the reflection coefficient have different values. The reason of my interest in obtaining the reflection coefficient from the reflected field is to compute the modal components of the reflection coefficient. Since adding transmission/reflection request to the characteristic mode configuration is not supported, utilizing the information of the modal reflected fields seems like the only option for the computation of the modal reflection coefficients. Besides that, although characteristic mode analysis is supported for models with periodic boundary conditions in FEKO 7.0, it is not supported anymore in FEKO 14.0, unless I am missing something. A similar work has also been done in http://ieeexplore.ieee.org/document/7695921/.
Thanks.
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Hello hykr,
Support for PBC and CMA was removed - it was not supposed to have been supported in the past. I have opened and issue to review this decision so that we can have a look at possibly adding support for this in the future.
For the transmission / reflection calculation, a phase reference position is specified. If you want to match the phase TR coefficients calculated from the near field calculations with those calculated directly by FEKO, you will have to ensure that both calculations use the same reference position. It is rather difficult to help you further without a model. You can send us your model if you would like us to help you with the phase.
Regards
JIF
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Hi JIF,
I have attached a model in which near field and the reflection coefficient are calculated at a distance of 10 wavelengths. Although the round trip phase shift should also be a multiple of 2*pi, there is a 180 degrees phase difference between the near field and the reflection coefficient results. I would greatly appreciate it if you explain its reason.
Thanks.
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Hi hykr
The transmission and reflection coefficients are calculated using the magnitude and polarization of the incident plane wave as reference.
You will notice that your incident plane wave has the incident E-field oriented (polarized) in the negative X direction - indicated by the red arrow in CADFEKO.
The near field request - by definition - is referenced in the positive X direction.
This results in the 180 degree difference between the reflection coefficient and the scattered near field.For your (scattered) near field to directly relate to the reflection coefficient the incident E-field would have to be polarized in the positive X direction (using a polarization angle of 180 degrees in the plane wave definition).
Regards,
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Altair Employee